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Theorem eldmcoss 35229
 Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
Assertion
Ref Expression
eldmcoss (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcoss
StepHypRef Expression
1 dmcoss3 35224 . . 3 dom ≀ 𝑅 = dom 𝑅
21eleq2i 2874 . 2 (𝐴 ∈ dom ≀ 𝑅𝐴 ∈ dom 𝑅)
3 eldmcnv 35134 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
42, 3syl5bb 284 1 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∃wex 1761   ∈ wcel 2081   class class class wbr 4962  ◡ccnv 5442  dom cdm 5443   ≀ ccoss 34985 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-coss 35190 This theorem is referenced by:  eldmcoss2  35230  eldm1cossres  35231
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